where poll n means that pa divides n, but pa + 1 does not. The importance of the abovefunctions comes from the fact that they represent partitions of n into sums of primesor prime powers which divide n, and recently several results concerning thesefunctions have appeared (see [1], [2], [4], [5] and [6]) . Thus it was proved in [2], eq .(5 .33), that one has

P (n)/P (n) = x+O (x log log x/log x),2-nix

.Y B(ii)/P(n) = x+O (x log log x/log x),2-n=x

and [4] contains a proof of

(1 .2)

Z B(n)/fl(n) = x+0(xexp(-C(logx-loglogx) 1` 2)), C > 0,2~n-x

and the same asymptotic formula holds for sums of f3(n)IB(n) . Sharp formulas forsums of reciprocals of P(n), f3(n) and B(n) are obtained in [6], where it was shown

1980 Mathematics Subject Classification . Primary 10H25 .Key words and phrases . Largest prime factor of an integer, additive functions, number of in-

tegers not exceeding x all of whose prime factors do not exceed y, asymptotic formulas .

The second author's research has been supported by Rep . Zaj . and Mathematical Institute ofBelgrade .

ESTIMATES FOR SUMS INVOLVING THE LARGESTPRIME FACTOR OF AN INTEGER AND CERTAIN RELATED

ADDITIVE FUNCTIONS

by

PAUL ERDŐS and ALEKSANDAR IVIe

Abstract

Let P(n) denote the largest prime factor of an integer n=2, and let

fl (n) = Z P , B (n) = Zap, B, (n) = Zp a .Pln

Palln

Palln

Asymptotic formulas for stuns of quotients of these functions are derived . The estimates are made todepend on yr (x, y), the number of integers not exceeding x, all of whose prime factors do not exceed y .

1 . Introduction

Let P(n) denote the largest prime factor of an integer n-2, and let us defineadditive functions fi(n), B(n) and B,(n) as

f3 (n) =

p, B(n) =

AP, B,(n) =

pa ,pill

Palin

pallo

Studia Scientiarum Vaíhematicarum Hungarica 15 (1980)

1 8 4

P. ERDŐS AND A. IVBC

that

(1 .3)

Z 1/P(n) = x exp{(-2 log x . log log x)112 +0 ((log x . log log log x)i 12)},2~n-x

and the same formula holds for sums of 1/fl(n) and 1/B(n) .Sums of quotients like those appearing in (1 .1) or (1 .2) provide us with informa-

tion about the degree of compositeness of n, and it turns out (see [1]) that it is P(n)which dominates the values of fl(n) and B(n) . Our Lemma 4 shows that the same isalso true for B1(n) . The main goal of our paper is to give estimates for the twelvedistinct sums of the type Z f(n)/g(n) when fig and

2-n =x

f, gE{P(n), /3(n), B(n), B,(n)} .

Estimates for some of these sums are already provided by (1.1) and (1 .2) andsome follow easily hencefrom, but a number of these estimates are non-trivial andwill be given in theorems of this paper .

The notation used throughout the text is standard : p and q are always primes ;in, n, r, s are natural numbers ; ~ (x, y)=

1 ; f=O(g) and f<<g both»<x,P(n)~y

mean If (x) Cg (x) for some C > 0 and x--xo ; C, C,, . . . denote positive ab-solute constants, not necessarily the same ones ; e denotes a positive number whichmay be chosen arbitrarily small . The notation /3(n)=Zp and B(n)= Zap stresses

PIn

pallnthe analogy between the relation of "large" additive functions fl(n) and B(n) andthe relation between the well-known "small" additive functions co(n)=,Zl and

Q(n)= Z a. Moreover, fl(n) may be regarded as the additive analogue of the mul-p°lin

tiplicative function a(n)= jjp, which represents the greatest square-free divisorPin

of n .

THEOREM 1 .

(2 . t)

Z P(n)/B,(n) = x+0(xloglogx/logx),2~nsx

(2.2)

Z Q (n)/B, (n) = x+ O (x log log x/log x),2sn~x

(2.3)

Z B(n)1B,(n) = x+0(xloglogx/logx) .2~n=x

Let now y denote Euler's constant, so that el =1 .78107 . . . , and let D > 1denote an absolute constant whose genesis will be precisely given in § 4 . We havethen

THEOREM 2 .

2. Statement of results

(2.4)

Z B, (n)/P(n) = elx log log x+O (x) .2_n~x

Studia Scientiarum Mathematicarunx Hungarica 15 (1980)

THEOREM 3 .

ESTIMATES FOR SUMS

1 85

B, (n)/B(n) = Dx+O(x log -113x) .2~n--x

THEOREM 4 .(2 .6)

B, (n)/Í3(n) = e x log log x + O (x) .2-nix

Therefore it still remains to estimate two of the twelve sums

f (n)/g(n)2-n-x

that are mentioned in the Introduction . These are

(2 .7)

P (n)/Í3 (n) = x+0 (x log log x/log x),2==n-"-x

and

(2 .8)

Z P(n)1B(n) = x+0(xloglogx/logx) .2-n--x

To obtain (2.7) note that the sum in question is trivially ~x and from (1 .1)and the Cauchy-Schwarz inequality we infer

x+0(1) _

1

(

P(n)/Q(n))1/2(

fl(n)1P(n))I122 Pt~x

2-n- -x

z-n<-.x

(

P(n)l/l (n)) I12(x+0(x log log x/log x))2<n-x

whenceP(n)//i (n) - x+0 (x log, log x1log x) .

2-n-x

This gives (2 .7), and (2.8) is proved analogously . The error term in (1 .1) (andconsequently in (2.7) and (2.8)) can be improved to 0(xllog x), which will be shownat the end of § 4 .

3. The necessary lemmas

We begin the preparation for proofs of our theorems by proving several lemmasof which some seem to be interesting in themselves . Our proofs will be made todepend on estimates for O(x,y), the number of positive integers not exceeding x,all of whose prime factors do not exceed y . From the wealth of results concerning0(x, y) we shall need several estimates whose proofs are to be found in DE BRUIJN[3] (to see that (3.2) holds for logy>IogSjS+Ex one has to use the strongest formof the prime number theorem), and apart from Lemma 1 below our proofs areself-contained .

LEMMA 1 . Let y--x and a=log x/logy . If 3-u-4y 1/'/logy, then thereexist constants c I , c 2 >0 such that

(3.1)

t (x, y) - cI x log' y • exp(-u (log u+ log log u-c 2)) .

If o(u) is defined as Q(u)=1 for 0----u--l, uO'(u)--Q(u-1) for u--l, thenfor logy >log' 111 -x ive have

(3 .2)

(x, y) = x o (at) (1 + 0 (log log x/logy)) .

Studia Scientiarum 1fathematicarum Ilungarica 15 (1980)

1 8 6

If s-0, x then

(3.3)

(x, y) << x/S !,

and for 2-y-x and some C>0

(3.4)

0(x, y) << x exp (- C log x/log y) .

LEMMA 2. Let S(x) denote the number of integers n-x such that P 2(n)jn .Then for some C > 0

(3.5)

S(x) << x exp (- C (log x • log log x)112)

PROOF . We have

(3 .6)

S(x) _

1 =

~(xp-2 , p) = Sl+S2 ,p2rnsx,P(m)=p

p2-x

where in S, we have p >exp ((log x • log log x)112)=z, and in S 2 we have p-z.We have

(3 .7) S, < 'Z Z t < x

-2 « x exp(- (log x • log log x) 112 ) .per= m`xp - 2

p i

For S 2 we use (3.1) to obtain with up =(logxp -2)/loge, C1 >0

S2 = Z ~ (xp -2 , p) « x log 2x'Z p - 2 exp(-Cl op log up) <~ Z

-Z-2 exp(- C(log x • log log x)i 12) << x exp(- C(log x • log log x)112)

since Up (10(y xz-2)/log z»(log x/log log x)112 for p--z .

LEMMA 3 . Let T(x) denote the number of integers n-x such that there existsq°In, q-P(n), q prime, for which qa>P(n) log -A x, where A>0 is arbitrary butfixed. Then

(3 .9)

T(.x) < (x log log x)/log x .

PROOF . With y=(log x •log log x) 112 we have ~ (x, exp y)<<x exp (-Cy) by(3.1), so we have only to consider those n-x for which P(n)>exp y, P(n)Ijn(this last by Lemma 2) . Therefore

(3 .10)

T(x) << x exp (-Cy)+T,(x)+T2(x),

where

T, (x) _

Zmgp~x, plog Ax<g p, P(m)<p

and if n is counted by T,(x), then there is a prime power gaIn, a72 such thatqa>P(n) log-Ax :~-exp (y/2) . Therefore(3.11)

7 2 (x) < x

n < x

11 -2 « x/log x.n, a~2, n"-exp(Y/2)

n- exp(Y/4)

P . ERDŐS AND A. IV16

(3 .8)

p

P

Stuiha. Sctentiarum Matlaemaúcarum Hungarica 15 (1980)

(3.12)

so that

(3.13)

To estimate Tl(x) we use (3.4) and

1/p = log (log B/log A) + O (l /log A), A BA<p~B

Ti (x) `

(x/pq, p)p x plog-Ax-q-p

I

z

1exp -Clogx/pq <<

p~ p plog -Ax<q<_p q

loge

<<x p- p

exp (- C log x/logp) • lo lolog x«

gp

<< x f loglogx exp(-Clogx/logt)d7r(t) <<2

tlogt

ESTIMATES FOR SUMS

lo xl

log log x logx/log2« x log log x exp

-C lo t li t - ' log -2 tdt < x

lo

e -c"du «2

g

g

I

187

log log xx --

,log x

after substituting u=log x/log t . The lemma follows then from (3 .10), (3 .11), (3 .13) .

LEMMA 4 . Let U(x) denote the number of integers n-x for which

B, (n) = P(n)(1 +0(loglog n/log n))

does not hold. Then

(3 .14)

U(x) << (x log log x)/log x .

PROOF . Let for brevity g(x) =log log x/log x . From Lemma 2 and Lemma 3 itis seen that for +0(.vg(x)) integers n ---x we have P(n)JIn and q°-P(n) log -3xif q°In, q-P(n), so that for these n's

B,(ra) = P(n)

~;

q° - P(n)+0)c rl)P(n)1og -3xq°11n, q -P(n)

P(,,)(1 +O(IOg 2x)) = P( ,, )( 1 . 0(g(n))) .

Since B,(n)=P(n) we have B,(n)=P(n)(l+0(g(n))) for x+O(xg(x))integers n_x, hence the lemma .

LEMMA 5 . The assertion of Lemma 4 ren;ain.', true when B,(n) is replaced byfl (n) or B(n) .

PROOF . Follows from the proof of Lemma 4 and

P(n) P) (r) = ú p = _, ap == B(n) =

B, (17) .pin

J"Iln

p°II

Studia Scientiarum Mathenaaticarum Hungarica 15 (1980)

188

P . ERDŐS AND a . wi(_

LEMMA 6 . lf'f(n) is any additire function and y=x, then

(3.15)

Z f(n) _

2' (f(Po)-f(pn-1))~(XP-a,Y) .n-=-x, P(n)`y

P" =x, p=y

PROOF .

(4.3)

f(n) _

f(pa) _

.Í(P") _x, P(n)`y

n=x, P(n)_y p°Iln

P°m~x . (P, n9=1 , p=Y, P(m)`-Y

_

(f(Pn)-f(p" -1))

1 =

2;

(f (Pn ) -f (P°-'»iP(xp-a, Y) .P« --x, P--y

rn xp -° ^111)- Y

p ° =x, P--Y

4, roofs of theorems

Theorem 1 (and also (IA)) follows easily from Lemma 4 and Lemma 5 . To prove(2.1) note that P(n)--B7(n), so that using Lemma 4 one obtains

n P(n)lB7 (n) = o(U(x))+

P(n)2s

P(n)(i+0(log log n/log n)) _nsx

2= nix

O(xloglogx/logx)+ Z (1+O(loglogn/log n))=x+O(xlog log x/log x),2 ,, x

and similarly one derives (2 .2) and (2 .3) .The proofs of the remaining theorems are more difficult and will be carried out

in three steps. The first step consists in proving

B,(n)/P(n) _

pr-1+O(x),2_-n-x

prmsx, P(rn)-p

(4.2)

B,(n)/B(n)=

z

r -ipr-1 +0(xlog-,/ax),2~n x

prm=x, P(,,,)-p

B,(n)IÍi(n) _

2Y

pr-1 +0(x) .

2-n~x

prrn=x, P(m)-p

The sums on the right-hand sides of the above formulas will be transformed intosums involving the function o(u) of Lemma 1, and the second step of the proof willbe to show that

(4 .4)

pr-1 = x

p-1

p

log x

logxI+s1+O(x),_Y ~ I

[p r m`x, P(n,) P

P_x

s=0

log p

log pp

(4.5)

Z

r -1pr-1 = x

1p"msx,P(m)-P

p=x p 0sS-Iogx/IOgP-I

[logx/logp] - S

0 (x log-1 / 3 x)

Studw Scientiarum Mathematicaram Hungarica 15 (1980)

log x

logxloge

logp +s

Finally it remains to simplify the expressions containing p(u), and our resultswill then follow from

(4 .7)

where

(4.8)

.2 PP-x

1 `° ~(logxI

_ logP

P ó` Iog p

log p +s = e'lloglogx+0(1),

lo Y

lo x ls

plog p - ~ log p J

= D+O ( 1/log x)0 --_5<-Iogx/,ogP_I

[log x/logp] - s

0-s v

After sketching this plan of our proofs, we begin with the proof of (4 .1) . Byadditivity of B,(n) we have

(4.9)

BL (n)/P(n) _

P"_1+

.2~

Bi(n2)lh,2~n- x

and (4.1) follows from

(4.10)

S = B1(m)1P « x .Prm-.x, P(M) -P

Noting that Bl(p°)=p° and usin-- Lemma 6 we obtain

(4.11)S=

p -r

B, (m)

~(xp -rq- ", p)q' .P11x

m=x/Pr , P(m)<P

P

qs`xlP", q`P

For some integer k-1 we have

(4 .12)

x/p'_" < q s

x/p ' _~ F-i ,

which implies

(4.13)

log x/p'` +r

slog q

logx/pk+r+log p,

so that there are at most

ESTIMATES FOR SUMS

2(u-[11]+s) du > 1 .1

[u]-s

P"m~x, P(m) -P

P•m~-x, P(m)-P

(4 .1.4)

log p/log q+ 1= 2 log p/log q

values of s for which (4.12) holds. Moreover, if (4 .12) holds, then we can use (3 .3)of Lemma I to obtain

(4.15)

i (xp-rq - s', p) « xl(prq's(k-1)!),

,

Studza Seientiarum Mathematicarum, Hun,garica 15 (~)

189

1 90

P. ERD(S AND A . IVIC

and therefore

S« 'Z

Zx

pS « Z XP-1 Z 1«pr-x gs'x/pr, q---p k 1 prg s (k-1)! P Pr-x

g---x/pr, g=P

(4.16)

«

- r-1 Z fog p/log g « x

- r -1 1og pPd7r (t)

«

pr-

g-p

pr-x

2 log t

«x

p - r -1p log p log-2p < x

1 /log p p-r < x Z 1/(p log p) « x,pr =x

Pr=x

r=1

p=x

(4.17)

(4.22)

since the last sum converges . This proves (4.10), and therefore (4 .1) .We turn now to the proof of (4.2) . By additivity of B(n) and B 1 (n) we

B, (n)/B (n) _2sn'x

pr/(rp+B(ni»+

Z

B,(m)/(rp+B(ni)).prm~x, P(m)<P

prm'x, P(m)<P

First we show

(4.18)

Si-

B,(m)/(rp+B(m)) « xlog -1/3xXprm-x P(1 n)<p

Let now for brevity w=1og 1 /3x throughout the proof of (4.2) . In the above sumwe may suppose p-exp iv, for otherwise following the reasoning given in (4 .16)we obtain

B,(m)/(rp+B(ni» «

p -1B,(m) «prm=x, P(m)<P, P>expw

prns-x, P(m)<P, P~exp~+'(4.19)

« x 757 1/(p log p) « x/w = x log-1 /3 x,P > exp w

since by the prime number theorem and integration by parts we have, as y-~,

(4.20)

Z l/(p log p) << ' t-1 log-1 t d7r (t) « 1/log y .p~Y

y

Next we observe that 1/(rp+B(m))-min (1/rp, I/B(m)) and

(4.21)

B,(n)IB(n) _ ( Z pn)I( Z ap)

a -1 pq-1 =f(n),p a ll,,

pa lin

p a ll,,

so that f(n) is additive . Therefore by Lemma 6

prm'x P(m)<p_'expw

~

1

1

s

x!a

(

p«p-x, P~exp w gs-xp-r, g-p

min ~pr sg g l pr gs,

St.i-ia Scientmrum Mathematiearum Hungarica 15 (1980)

B, (m) <Gr•p +B(nt)

have

This is the fundamental inequality in the proof of (4.2) . Denoting by f theexpression on the right-hand side of (4 .22) we may write

(4.23) Z -

+ Z +

Z

_ Z1 -PZ2 + ,Z3>rts ~-logx/(21ogp) -logx/(41ogp)

s- log x/(4 log p)

since if r+s>log x/(2 log p), then either r>log .x/(4 log p) or s= log x/(4 log p) .In E1 we have

(4.24) log xp - r q- s/log p - log x/log p-r-s - log x/(2 log p) -_ log x/(2w),

whence by (3.4)

(4.25)

O (xp- r q- , p) « xp-rq-S exp (-Clog x/w),

and since trivially s«log x we obtain then

(4.26),Z1 « x exp (-C log x/logs/3r) Z p-r_1 log x Z 1 « x exp (-Cl log'/3 x) .P ,•_ x

o __ P

Now we come to the estimation of E 2 and E 3 in (4 .23) . In E 2 it is seen that r islarge, so that we shall take min (1/rp, 1/sq)-1/rp, and in E3 we shall takemin (1/rp, 1/sq)-1/sq . In the estimation of E 2 and E3 we repeat the reasoning givenby (4.12)-(4 .16), taking also into account that p=exp w . Since in E 2 we have1/r«loge/log x, we obtain

(4.27)

(4.28)

«x

~z p-r « x log -1 x Z 1/p « x log log x •log -lx .log x P_exp w r-1

P-exp w

Similarly for E3 we obtain analogously as in (4 .16)

ESTIMATES FOR SUMS

1 9 1

X

'Z.Z2r>logx/( 4 1ogp) « lOg x Pr=x, p expw

p r 1 log'p

1/(lOg q) «

2Y3 =s> log x/( log P)

Slgs-'1

x« [ p q,> p «P"`-= x, P-expw 9'~~xP r, 4=A s>logxí(4logP)

« x log -1x _Y 1/(k-1)!

_Y

p-r log p

Z

1/q «k~1

P--x, P-expw

NS =̀'xP - ', 9=P

« x log-1x

p-r log''p L 1/(q log q) «P.-x, P-exp w

9`-P

exp w

« x log-lx

lcg2p .2 p-r « x log-lx f t-1log 2 t d7r (t) «P-exp w

r=1

2

« xwz log- lx = x lag-1 / 3 x,

since by (4.14) there are 0 (log p/log y) choices for s .

Studies Seientiarum iMaíhemaűcarum Hungarica 15 (1980)

1 9 2

We have shown that (4 .18) holds, and to finish the proof of (4 .2) it remains toprove

(4.29)

prl(rp+B(m)) _

r -1pr-1 - O(xiog -13 x),P rmjx . P(^OAP

P rm`x, P(M ) -- P

which after subtraction follows from

(4.30) S, -

The sum S2 is easier to estimate than S, of (4.18), and by Lemma 6 we have

(4.31)

S-,«

,-2 pr_2

-

q~(xp`q`, p),_

since B(p°)-B(p 1)=ap-(a-1)p . We estimate first the subsum -Y' of the sumon the right-hand side of (4 .31) with p>exp w, w=1og 1 /3x. Trivially we have

f

I.-2pr_2

Z

qxp -r q ,pr `'x, p~expw

qs-xp q` P

(4.32) « x

) -2 p-2

q - i « x

Z

r-2 p -I log -ip «P' -x . P>CxP-

qxp j-0

P-x P>expw

« x Z 1 .-2 Z 1/(p log p) « x/w = x log -1 /3x,r~l

P-CxP w

where we have used again (4.20) . In the remaining subsum E" we have p-exp w,and we split it analogously as in the case of S, in (4.18), i.e .

(4 .33)

+

Z

+

Z

= ,Z1 Z2+ r-{-s logx/(21oüP)r7 log x/(4 log p) s>logx((41ogp)As for 2:, of' (4.23) we obtain similarly

(4.34)

« x exp (- C log=/3x) .

Since r«log x we further have

Z

« log-2x

P. ERDŐS AND A. IVIC

)' -2Pr_

2 B ( y)) «xlog-1 aaP r n,- x, P ( ,,, ) - P

pr-21og2p

Z

qxp- rq «r>logx/(41ogP)

P " =x, P expw

gs -xp - ", q"~ P

pr, x, p .-exp ti,

Finally we have

(4.36) Z3 =

Z

«

r-2 pr-2

gxp-r q-' «s>logx/(4logp)

P" -X, P-`expw

q sqxp -r, q --- p, s'~logx/(41ogp)«x-2_lo1"")

Z

p -Ir -2 Zlogx«xexp( - Clog 2 / 3 x),

(4.35)

«x log- 2X

p-slogs z) Z Z q-' <<

X log_sx

>'

p -rlog p « x log - " x • log x •l og (exp w) = x log -2 /3x

PI-x, P=exp w qsp

since in I" we have q -' 2 -'°gx ( ''w) q--p and s«log, . Therefore we have proved(4.30), completing the proof of (4 .2) .

StucLia Scienti rum Mailwmatwarum Hun.gariea 15 (1980)

Up to now we have proved (4.1) and (4.2), and now we move to the proof of(4 .3), viz .

B, (n)/f3 (n) _

Z

pr-1 +O(x)-2~n`x

Prm-x, P(m)<P

By additivity of BI(n) and fl (n) we have

(4.37) Z BI(n)/P(n) _

Z

prl(p+a(m))+

BI(m)/(p+a(m)) .2-nor

prm~x, P(m)<p

prm-x, P(m)<P

By (4 .10) the last sum above is O(x), and so it remains to show

(4.38)

Z

(pr -I-pr/(p+ fl(M))) «

57

pr -2f (m) « x .Prm-x, P(m)-p

prn:-x, P(m)<P

The first inequality in (4 .38) is obvious, and for the second we note that#(pa)-fl(pa-1)=p-1 for a=1 and zero for awl, so that Lemma 6 gives

(4.39)

ESTIMATES FOR SUMS

pr-1f(m)« pr-2 -Y q, (xp-rq-1, p) .

Prm-x, P(,1,) -P

Using (3 .4) we have

i (XP-rq-1 ,p) «xp-rq-1 exp(-Clogx/(log p)+Cr),

which gives then

(4.40)

pr -2 f3(ni) «x Z p-2 ~' exp -C . l og x +Crl «prrn -x, P(m)~ p

Pr=x

q-p

log p

« x

(p log p) -' exp (- C log x/log p) Z exp (Cr) « x Z 1/(p log p) « x,P-x

r=logx1logp

Psx

as asserted .Now we shall pass to the proof of (4.4) and (4 .5), but first we need to clear a

technical point . Since the function 0 (x, y) is defined as

we remark that

(4.41)

(4.42) Z

r -1 pr-1 =

Kx, Y) _ 2Y 1,n=x, P(n)-y

pr-I =

Z

pr-1 +0(x),prm-x, P(m)<P

prm-x, P(M)--P

prm-x, P(m)-P

P-M-x' P(M)--PZ

r-1 pr- ' + O (x log log x/log x),

which will facilitate later transformations of our sums . To obtain (4 .41) note that

S'

pr-1 -

pr-I =

Z

Pr-1 =

Pr n+-x, P(M)`P

Prm` 'P( )<P

p111,-x, P(n) 5p

xpr-1,/,px p

Pr+1sx

193

13

Studio Scientiarum Mathematicarum Hungarica 15 (1980)

194

since if P(m)=p, then m=np with P(n)-p . With (3.4) we obtain

(4.44)

(4.47)

Z Pr- ' ~ (xp -r - ' , P) < xpr .~ I5X

Pr

P. ERDőS AND A . IV16

p-2exp(- Clogx/logp)exp(Cr) <<I x

<< x Z p-2 exp (-C log x/log p) Z exp (Cr) << x Z p -2 << x,Psx

n--logx/IogP

Pox

and the proof of (4.42) is analogous, when we consider separately the casesr - log x/(2 log p) and r > log xl (2 log p) .

IFo prove (4 .4) note first that we may take r-2 in (4 .1), since the sum withr=1 is trivially O(x) . Suppose now a=[logx/loge], or equivalently

(4 .43)

pa = x -- Pa+] .

Writing r=a-s we have s=0 1 1, . . .,a-2, so that s can take at most O(log x)values. Therefore we can write

Z =

Z

pr-1 =

pa-s-I~(xps-a , p) .prmsx, P(m)~p r?2

pa-s~x

If s < Jog"' p then

(4.45) a=Jog xps- °/loge=log x/loge-[log xllog p]+s<s+l <2 log'/2 p,

so that for s< 10g'/2p we may use the asymptotic formula (3.2) to estimateO(xp' - a, p) in (4 .4) . Writing

(4.46)

= Z +

= Z) +Z2,stlogl 12P

s=alogv 2p

we obtain then

1

log x

logx~, = x ~ P

N

+ s -1-psx

0_s-min(logx/IogP, logli 2 p)

log P

log P

+0(x Z P-1

Z

log-1 p •log log (xp'- a) . O(S».PIx

o,-s_logI 1 2p

But in view of (4 .43) log log (xp' -a)«log log (s+ 1)p, and from the definingproperties of e(u) it is seen that o(u) is nonincreasing and that P(s)«7/s!, whichgives for the error term in (4 .47)

X

(4.48) O (x Z p _' log-1 p • l og log p) = O (x f t - I log- I t •log log tdn (t)) =O (x),PIx

y

after integrating by parts and using the prime number theorem . Next we have

_~

log X _ ( log xZ P

Z P

L

,+s = ZP-,

Z 0(s)<<(4.49) Psx

s?logx/logP

loge

IogP

P-x

s1logx/logP

<< Z p-' Z 1/s! << log -'x Z P-1 log p< lP=x

s.--Iogx/IogP

Psx

Studia SeíenUarum Mathematicarum Hungarica 15 (1980)

ESTIMATES FOR SUMS

and similarly

(4.50)

This implies

(4.51)

1 = x

P-I

(logx -[logx l+s) { O(x) .

P._x

s=o

log p

log Jp

Using (3.3) we obtain

G2 = Z Pa- .'-11/' (xP •s -a ' P) « x Z 1./P Z t/S!s--]0g 1/2p

pa—X

s-10g112 p« x

p-1 exp (-logl 2 p) « x -'-'I l(plog(4.52)

which finishes the proof of (4.4) in view of (4.41), i .e .

(4 .53)

(4 .56)

pr =p"m-x, P(m)- P

zP I

Q(logx_[logxl+s) « 1 .Pox

.=1ogi/2p

log P

log PJ

P,X

P

psx

.2

r_i Pr-1 =

-s) - 1pa- .s-11p(xps

p) _P rm`x, P(^)`P

2Y +

2

= SI. i-S2 .p=exp(Iog2iax)

exp(j0g213 x)~p X

We observe that for u-_l we have [u]-_u/2, so that

(4.54)

1l((a - s)s!) « t/a,0-.s a_1

and with a=[logx/loge] we obtain using (3.3)

(4 .55)

S1 «xlog -, x

p -1 logp«xlog -1/ 3x .P'cxp (1og 2 ' 3 x)

For S2 in (4.53) we have s < log x/loge < logI 11p, and so as in the proof of (4.4)we may use the asymptotic formula (3.2) to evaluate 0(xps-a, p) . Therefore

(logxS2 =-x

X

p1

0 loexp(IOg2i 3 x)<p-X

0=s_logx/logp-1

g P

log x

logx~ (logP [logpI+s +O(x) .

195

'1 'o prove (4 .5) we shall need (4.42), and writing again r=a-s, a=[log x/loge],we obtain

[logp] +s ) ([ loglogxPI-s) 1 1

log log xps-a (log x logx )([loge

13'

Studia Scienliarum Mathematicarum Hungarica 15 (1980)

196

As in the proof of (4.4) we have log log (xps -n)«log log (s+2)p, so that theerror term above is

(4.57)

O (x log -IX 'Z p-1 log log p) = O (x (log log x) 2/log x) .P_x

Using (4.54) and o(s)«1/s! we have

x

-,lobg x

logx

l log PI ))1og

x

l I(4.58)

PPm5exp(log2 ' 3 x)

oI logp

-[ logp

l]+sl~[- -sl

«xlog -1 /3x,

which completes the proof of (4.5) .Now we finally come to the simplification of sums which appear in (4.4) and

(4.5) and involve Q (u) . If one wants only to show that the sums in (4.6) and (4.7)are asymptotically equal to C log log x and C, respectively, with ineffective C's andwithout error terms, this can be obtained by elementary methods using only thecontinuity of o(u) . To prove (4.6), however, we shall use the prime number theoremand Stieltjes integral representation to obtain

~

1 `~ ~(logx (logxl

_Z p

s~ log p log p +s

logx

logxl +sl drz t

Y_0

0 log t - [ log t l

( )(4.59)

since o(u)<<e - n, which follows from uo'(u)=-o(u-1) . The second integral aboveis 0(1), and in the first integral we make the change of variable u=log x/log t andobtain

GS

I

~ log xI

log xIpsx p só o log p

log p +s

log x/log 2f

u-Io(u-[u]+s)du+0(1) .s~ I

We transform the series on the right-hand side above as follows :logx/log2

f u-Io(u-[u]+s)du =s-0 I

(4.60)

(4.61)

P . ERDŐS AND A. IVIŐ

( log x-L

logxJ

dt

' t -Ids 2

log t

log t+s

tlogt+O~se_s f

(O(tlog-~t))),

logx/log2= loglogx+0(1)+

o(s) f u- Idu+s=1

1

logx/log2

+

f

u-1(o(u-[u]+s)-o(s))du =S=1 1

n+l u-(u]+s

_

Q(S)Ioglogx+0(1)- Z

Z

f U- I f t -Io(t-1)dtdu,ss=0

s=1 n_logx/log2 n

Studia Seientiarum Mathematicarum Hungarica 15 (1930)

x

where we have used o(u)=1- f ~o (t-1)t-Idt, which follows fromI

uó (u) _ -g(u-1) . Changing the order of integration gives

(4.62)

n+1

u-[u1+s

s+1

n+I

f u -1 f t -1 o(t-1)dtdu = f t -1 O(t-1) f u -1du =•

s s t+n-s

s+1

(

S + 1 - t=

t-1 Q(t-1)log 11+ dts l

t-f-n-s

s}1

s+1-t= f t -1e(t- 1)dt+O(e- sn-2) _

s t+n-s

s+1=n-1f o(t-1)t -1(s+l-t)dt+0(e - sn -2 ) .

s

Therefore from (4 .61) and (4 .62) we obtain

ESTIMATES FOR SUMS

1 9 7

n+1

u-[u[+s

f u-1 f t-12(t-1)dtdu =s=1 n~logxllog2 n s

s+1

_

(log logx+0(1)) f t -12(t-1)(s+i-t)dt+0(1) _s

s+1

- s+I

(4.63) = log logx(z (s+1) f t -1 o(t-1)dt-

f o(t-1)dt)+0(1)s

_s

•

s

_ - f o(t)dt .loglogx+0(1)+loglogx~(•

s+1)(O(s)-e(s+1))=o

_ - f o(t)dt •loglogx+0(1)+loglogx(l+

Q(s)) _o

s=1

_ (~ e(s)- f e (t) dt) log log x + 0 (1) .s

Putting (4 .63) into (4 .61) we obtain (4.6), which completes then the proof ofTheorem 2 and Theorem 4 . It remains yet to prove (4 .7), which will give then

Studia Scíentiarum Mathematicarum Hungarica 15 (1980)

198

Theorem 3. We have similarly as in the proof of (4.6)

P1

(logx~ +s ([log x/log P]--s)P

-1 -O ~ slogxllogP-1

ologe

logxloge

l

(4.64)

(4.67)

J

1

(logx [log x]1

(

1 -_1- d~(t)

o

ls [Iogxüogt]-s) -2_0

o~s<,ogx,ogt-1

log t

log tx

to x [lo' vl= J t-,. log - ' t0-s-'ogx/togt 1 ~ log 1

log t +s ([log x/log t]-s)--ldt-{-

x log tO

t log x d (O (t log -2 t))) _

logxllog2

u -1

o(u-[u]+s)([u]-s)-Sdu+0(l/logx) =2

O--s-u-1

P . ERDŐS AND A. Me

o(u-[u]+s)([u]-s)-1-du+0(

f u-2 du)+0(1/logx)O s3u-~

Iogx/log2

= D+O(1/logx),

where D is given by (4.8) . This completes the proof of all of our theorems, when wenote (see [2], p. 314) that

(4.65)

io(t)dt = el = 11 .78107 . . . .0

In concluding we shall show how the error term in (1 .1), (2 .7) and (2 .8) can beimproved to O(x/log x). We shall only sketch the proof of

(4.66)

,Z /3(n)/P(n) = x+O(x/logx),2_n-x

since the proof of the analogous formula with B(n) in place of /3(n) is only technicallymore complicated . Using Lemma 2 we have

Z a(n)IP(n) _

Z

fl(n)lP(n)+O(

z

w(n)) _25nsx

2--n--x, P(n)Iln

2t 11=-X ' Pti(n) Jn

~3 (n)/P(n) + 0 (x exp ( - C (l og x • log l og x)i 12 )),2snsx, P(n)11n

since (3(n)-P(n)r)(n)<<P(n) log x for n- x . Further using Lemma 2 and Lemma 6we have

Z

P (11)/P(1l) _

(fi(P)+P(m))/P =2-n--x, P(n)Iln

Pmsx, P(m)<P

(4.68)

= X+Ox+ Z P-i

Z

P (m) _(log x

P_x

m~x/P, P(m){P

x-X+ O

( log x )

P--~x

, -~~+0(

P-1 eSxl 9- nqip (x/qP, p»,

Studía Scientiarum Nlathematicarurn Hun,garica 15 (1980)

since /3(yr)=y . With (3.4) and the prime number theorem we finally have

(4.69)

when we substitute it= log x/log 1 .

KATEDRA MATEMATIKERUDARSKO-GEOLOSKIFAKULTETUNIVERSITETA U BEOGRADUDJUSINA 7YU-i 1000 BEOGRADYUGOSLAVIA

ESTIMATES FOR SUMS

p-1 Z qIp (x/9p, p) «

xp-" Y exp (- C(]og x/9p)/log p) «

MTA MATEMATIKAI KUTATÓ INTÉZETEREÁLTANODA U. 13-15 .H-1053 DUDAPESTHUNGARY

X Z (p log p) - I exp (- C log x/log p)

x f (t log t)_I exp (- C log x/log t) - dhr(t) «

-n.C

«x J t-Ilog - '2 t •exp(-Clogx/logt)dt=

logxIlog2= x J

log - 'x •exp(-Cu)du «x/log x,

REFERENCES

[1] ALLADI, K . and ERDŐS, P ., On an additive arithmetic function, Pacific J. Math . 71 (1977), 275-294. MR 56 # 5401 .

[2] ALLADI, K ., and ERDős, P., On the asymptotic behavior of large prime factors of integers, PacificJ. Math . 82 (1979), 295-315 . NIR 81c : 10049 .

[3] BRUIJN, N . G . DE, On the number of positive integers - and free of prime factors >y, Nederl.Akad. Wetensch. Proc. Ser . A 54=1ndag . Math . 13 (1951), 50-60. MR 13-724 ;1 1 . Nederl . Akad. Wetensch . Proc . Ser . A 69=lndag. Math . 28 (1966), 239-247 .MR 34 # 5770 .

[4] DE KoNINCK, J.-M ., ERDős, P. and Ivte, A., Reciprocals of large additive functions, Canad.Math. Bull. 24 (1981), 225-231 . Zbl. 463 10032.

[5] DE KONINCk, J.-M . and lvié, A ., Topics in arithmetical functions, Notas de Matemática 72,North-Holland Publishing Co ., Amsterdam, 1980. MR 82a : 10047 .

[6] Ivj6, A., Sum of reciprocals of the largest prime factor of an integer, Arch. Math . (Basel)36 (1981), 57-61 . MR 82g : 10067 .

199

(Received April 8, 1981)